Question

plot all intercepts and asymptotes of the following rational function. if you get a fraction or decimal then plot as close to the true location as possible.
$f(x) = \frac{5}{x - 4} + 2$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept
click on a feature then drag it into place.

Explanation:

Step1: Find Vertical Asymptote

The vertical asymptote of a rational function occurs where the denominator is zero (and numerator is not zero). For \( f(x)=\frac{5}{x - 4}+2 \), set \( x - 4 = 0 \), so \( x = 4 \).

Step2: Find Horizontal Asymptote

For rational functions, when the degree of numerator and denominator (after simplifying) is considered. Here, the term \( \frac{5}{x - 4} \) has a horizontal asymptote \( y = 0 \) (since degree of numerator < degree of denominator), and adding 2 shifts it up, so horizontal asymptote is \( y = 2 \).

Step3: Find x - intercept

Set \( f(x)=0 \): \( 0=\frac{5}{x - 4}+2 \). Subtract 2: \( -2=\frac{5}{x - 4} \). Multiply both sides by \( x - 4 \): \( -2(x - 4)=5 \). Expand: \( -2x + 8 = 5 \). Subtract 8: \( -2x=-3 \), so \( x=\frac{3}{2}=1.5 \).

Step4: Find y - intercept

Set \( x = 0 \): \( f(0)=\frac{5}{0 - 4}+2=\frac{5}{-4}+2=\frac{-5 + 8}{4}=\frac{3}{4}=0.75 \).

Answer:

• Vertical Asymptote: \( x = 4 \)
• Horizontal Asymptote: \( y = 2 \)
• x - intercept: \( x=\frac{3}{2} \) (or \( 1.5 \))
• y - intercept: \( y=\frac{3}{4} \) (or \( 0.75 \))